I am studying a variant of the notion of (strict) n-categories, and I would like to show that this makes a cocomplete category. For that I planned on adapting the proof of the cocompleteness of n-categories but I cannot seem to find one.
One idea I had was to describe n-categories as algebras of a certain monad T over globular sets, but I ended up in a loop: the best way I can think of to describe T is by first describing the category of the n-glob (by hand), then by saying that the free n-category over a globular set consists in "gluing n-globs together". However, this gluing operation is formally performed by taking a coproduct, which we can't do since we are trying to prove that they exist!
So I am looking for a proof of the cocompleteness of nCat. Hopefully it would be a proof that does not rely too much on the exact structure of n-category so that I could adapt it to an other setting.